Determine the number of ways a jury of 12 can be selected from a pool of 20 people.
The Correct Answer and Explanation is:
To determine the number of ways a jury of 12 can be selected from a pool of 20 people, we can use the concept of combinations, which is represented mathematically as ( C(n, r) ) or ( \binom{n}{r} ). Here, ( n ) is the total number of people to choose from (in this case, 20), and ( r ) is the number of people to be selected (12).
The formula for combinations is given by:
[
C(n, r) = \frac{n!}{r!(n – r)!}
]
Where ( ! ) denotes factorial, which is the product of all positive integers up to that number.
Step-by-Step Calculation:
- Identify the values:
- ( n = 20 ) (total people)
- ( r = 12 ) (people to select)
- Plug in the values into the formula:
[
C(20, 12) = \frac{20!}{12!(20 – 12)!} = \frac{20!}{12! \cdot 8!}
] - Calculate the factorials:
- ( 20! = 20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12! )
- Thus, ( \frac{20!}{12!} = 20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 ).
- Simplify:
[
C(20, 12) = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{8!}
]
Where ( 8! = 40320 ). - Calculate the product of the numbers:
[
20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 = 125,970,720
] - Divide by ( 8! ):
[
C(20, 12) = \frac{125970720}{40320} = 3125
]
Conclusion:
Thus, the number of ways to select a jury of 12 from a pool of 20 people is 125970.
This calculation demonstrates the application of combinations in scenarios where the order of selection does not matter, which is typical in jury selection and similar problems. By using factorials and simplifying the equation, we find a large number of possible combinations, illustrating the diversity of groups that can be formed from a given set.